A066230 f-perfect numbers defined by f(n) = n - 1 (where f-perfect numbers are defined in A066218).

1, 12, 196, 368, 1696, 30848, 437745, 2075648, 8341504, 33452032, 34355150848

Offset

1,2

Comments

Equivalently, let g(n) = sigma(n)-n-d(n)+2, where d(n) is the number of divisors of n and sigma(n) is their sum. Then n is in the sequence if g(n)=n.

It seems that if 2^(i+1)-(2*i+1) is prime, 2^i*(2^(i+1)-(2*i+1)) is in the list. For example i in {2, 4, 5, 7, 10, 11, 12, 17, 24, 30, 34, 113, 151, 185}. No other exceptions than 1, 196 and 437745 for n < 10^8. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 31 2005

If 2^(i + 1)-(2i + 1) is prime then n = 2^i*(2^(i + 1)-(2i + 1)) is in the sequence because sigma(n)-d(n) + 2 = (2^(i + 1)-1)*(2^(i + 1)-2i)-2(i + 1) + 2 = 2^(i + 1)*(2^(i + 1)-(2i + 1)) = 2n, so sigma(n)-n-d(n) + 2 = n. -Farideh Firoozbakht, Sep 18 2006

a(12) > 2*10^11. - Donovan Johnson, Jun 25 2012

a(12) > 10^13. - Giovanni Resta, Aug 21 2013

Links

Table of n, a(n) for n=1..11.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Example

f(12) = 11 = 0 + 1 + 2 + 3 + 5 = f(1) + f(2) + f(3) + f(4) + f(6), hence 12 is a term of the sequence.

Mathematica

g[ n_ ] := DivisorSigma[ 1, n ]-n-DivisorSigma[ 0, n ]+2; For[ n=1, True, n++, If[ g[ n ]==n, Print[ n ] ] ]

Crossrefs

Cf. A066218, A066511, A066229.

Sequence in context: A296942 A159498 A034671 * A262845 A358363 A321033

Adjacent sequences: A066227 A066228 A066229 * A066231 A066232 A066233

Keyword

nonn,more

Author

Joseph L. Pe, Dec 18 2001

Extensions

Edited by Dean Hickerson, Jan 10, 2002.

More terms from Jason Earls, May 14 2002

2 more terms from Farideh Firoozbakht, Sep 18 2006

a(11) from Donovan Johnson, Jun 25 2012

Status

approved